The Regulator
Suppose that u1,...,ur are a set of generators for the unit group modulo roots of unity. If u is an algebraic number, write u1, ..., ur+1 for the different embeddings into R or C, and set Nj to 1, resp. 2 if corresponding embedding is real, resp. complex. Then the r by r + 1 matrix whose entries are has the property that the sum of any row is zero (because all units have norm 1, and the log of the norm is the sum of the entries of a row). This implies that the absolute value R of the determinant of the submatrix formed by deleting one column is independent of the column. The number R is called the regulator of the algebraic number field (it does not depend on the choice of generators ui). It measures the "density" of the units: if the regulator is small, this means that there are "lots" of units.
The regulator has the following geometric interpretation. The map taking a unit u to the vector with entries has image in the r-dimensional subspace of Rr+1 consisting of all vector whose entries have sum 0, and by Dirichlet's unit theorem the image is a lattice in this subspace. The volume of a fundamental domain of this lattice is R√(r+1).
The regulator of an algebraic number field of degree greater than 2 is usually quite cumbersome to calculate, though there are now computer algebra packages that can do it in many cases. It is usually much easier to calculate the product hR of the class number h and the regulator using the class number formula, and the main difficulty in calculating the class number of an algebraic number field is usually the calculation of the regulator.
Read more about this topic: Dirichlet's Unit Theorem